Proving the Trigonometric Identity: sin^2x / (1-cos x) 1 - cos x

In trigonometry, proving identities is a fundamental skill that enhances your understanding of the relationships between trigonometric functions. One such identity is the equation sin^2x / (1-cos x) 1 - cos x. This article will guide you through the steps to prove this identity, utilizing key trigonometric formulas and algebraic manipulations.

Step-by-Step Proof

Step 1: Simplify the Left-Hand Side (LHS)

LHS: sin^2x / (1-cos x) Multiply by 1-cosx/1-cosx: sin^2x(1-cosx)/((1-cosx)(1-cosx)) Utilizing the formula 1 - cos^2x sin^2x sin^2x(1-cosx)/sin^2x 1 - cosx

Step 2: Verify the Result

The LHS simplifies to 1 - cosx, which is equal to the Right-Hand Side (RHS).

Step 3: Verify LHS RHS

Left Hand Side: sin^2x / (1-cos x) Multiply the numerator and denominator by 1-cos x sin^2x(1-cos x) / (1 - cos^2x) Using the identity 1 - cos^2x sin^2x sin^2x(1-cos x) / sin^2x 1 - cos x

The LHS simplifies to 1 - cos x, which means LHS RHS proving the identity sin^2x / (1-cos x) 1 - cos x.

Key Takeaways

The identity sin^2x / (1-cos x) 1 - cos x is a fundamental relationship in trigonometry that can be proven through algebraic manipulation and the use of basic trigonometric identities. Understanding and practicing proving trigonometric identities will enhance your problem-solving skills and deepen your knowledge of trigonometry. Key steps include manipulating the fractions and utilizing the trigonometric identity 1 - cos^2x sin^2x to simplify the expression.

Now that you have seen the proof of this identity, you can use it in various mathematical and real-world applications. Keep practicing similar problems to refine your skills.

Related Trigonometric Identities:

sin^2x cos^2x 1 tan x sin x / cos x cot x cos x / sin x

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